Girih

How did medieval craftsmen lay out patterns of such staggering intricacy on the walls of mosques and shrines, without algebra and without measuring instruments beyond the compass and straightedge? For the earliest girih work, the classical tools suffice. But in 2007 the physicists Peter Lu and Paul Steinhardt argued in the journal Science that by around 1200 CE the designers had made a conceptual leap: they stopped drafting each star anew and began assembling patterns from a small kit of prefabricated shapes, which Lu and Steinhardt named girih tiles. There are five: a regular decagon, an elongated hexagon, a bowtie, a rhombus and a regular pentagon. All have edges of the same length, and every edge is crossed at its midpoint by decorating lines at fixed angles of 72 or 108 degrees. Fit the tiles together edge to edge and the decorating lines join automatically into a continuous strapwork, so a craftsman could build an enormous, flawless pattern the way a paver lays a floor, without ever computing an angle.

The masterpiece of the method is the Darb-i Imam shrine in Isfahan, dated 1453, where large girih tiles are subdivided into smaller copies of the same shapes. Carried on indefinitely, that subdivision produces a pattern that never repeats, a quasi-crystalline order of the kind Western mathematics first described with Penrose tilings in the 1970s. The claim is not that the builders held the modern theory, but that their craft system reached, in practice, a structure the West would need another five centuries to name.

Islamic architectural geometry left almost no treatises explaining itself, and for a long time that silence was filled with speculation. What it left instead were working documents, and the most important survivor is the Topkapi Scroll, a long roll of pattern drawings from the Timurid-Turkmen world of the fifteenth or sixteenth century, now in the Topkapi Palace Library in Istanbul.

The art historian Gulru Necipoglu, who published the scroll in full, showed what kind of document it is: not a book of doctrine but a designer's repertory, pattern after pattern drawn with the construction lines left visible. Under the inked strapwork run faint uninked lines, scored with a stylus, recording the polygonal scaffolding on which each knot was built. A workshop that owned such a scroll owned the grammar of girih: an apprentice could lift the construction, scale it to a wall or a dome, and adapt it to brick, stucco, wood or tile. Necipoglu drew a further conclusion from the working papers. The geometry was a craft literacy shared between architects, mathematicians and artisans, taught by demonstration and transmitted in scrolls like this one, rather than an esoteric symbolism handed down beside the patterns. Where meaning was attached to the work, it was attached by writers and patrons in particular times and places, not carried inside the knots themselves. The scroll, in other words, answers the how of girih precisely, and leaves the why to history.